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Power Allocation in Cooperative Networks Using

Differential Space-Time CodesBehrouz Maham and Are Hjrungnes

UniK University Graduate Center, University of Oslo, Oslo, NorwayEmail: behrouz@unik.no,arehj@unik.no

Abstract Estimating channel state information (CSI) in thefast fading conditions is very challenging. In this paper, a simplestructure for cooperative diversity in decode-and-forward modeis examined, in which the transmitter, the relay and the receiverdo not require to know CSI. In this scheme, transmissionof information is done in a two phase process. In the firstphase, differential modulated signals are radiated from thesource. After decoding the received signal in the relay side,the source and the relay collectively send information usingdifferential space-time codes. The optimum power allocationin a sense of minimizing pairwise error probability (PEP) inhigh SNR scenarios has been obtained using an analytical

method. An interesting property of the optimum point is that itis independent of the channel statistics and the position of therelay, which improves its feasibility in the future communicationsystems. Numerical simulations verify the analytical results.

Index Terms Cooperation, differential space-time codes,decode-and-forward, power control.

I. INTRODUCTION

Cooperative diversity allows a collection of radios to relay

signals for each other and effectively create a virtual antenna

array for combating multipath fading in wireless channels. A

majority of the work on cooperative diversity has focused on

scenarios in which the receivers, and perhaps the transmitters,

obtain channel state information (CSI) in the form of accurate

estimates of the fading coefficients (see, e.g., [1]-[5]). The

receivers can exploit available CSI for coherent reception, and

the transmitters can utilize available CSI for power control

and coherent beamforming. Estimating channel coefficients is

achieved by using the training sequences or blind methods,

but when channel conditions are affected by fast fading, the

estimation of CSI becomes inaccurate and impractical. For

the case that CSI is unknown in both the transmitters and

the receivers, some few schemes are presented. Recently,

the non-coherent detection is investigated in a relay-based

network with repetition codes [6]. Another way of dealing

with the issue is to employ differential modulation schemes.

Differential space-time coding has been considered in [7]-

[10] when the CSI is not available at either MIMO source

and destination terminals. The extension of this method to

the cooperative systems is considered in [11]. Furthermore,

[12] and [13] studied on differential modulation schemes in a

distributed manner with repetition-based codes. An introduced

scheme in [13] operates under BPSK. In addition to the

repetition codes, [12] studies the usage of differential unitary

This work was supported by the Research Council of Norway throughthe project 176773/S10 entitled "Optimized Heterogeneous Multiuser MIMONetworks OptiMO".

Fig. 1. Communication between source and destination by employing a relay.

space-time codes in a cooperative manner. But, instead of thedense orthogonal space-time codes which are studied in this

paper, it uses diagonal matrices as a unitary matrix. Moreover,

decoding at the destination side in [12] is just based on

transmitting signals from relays in phase II, which is not

efficient.

In this paper, we develop a distributed differential space-

time system based on [11], which uses Alamouti scheme as

a unitary matrix. Decoding in the destination is based on

the transmitting M-PSK differentially modulated signals in

both phases and the source node contributes in transmitting

signals in the second phase. Then, we analyze the pairwise

error probability (PEP) of the system. A simple power control

scheme based on minimization of PEP that is independentof the channel statistics and the position of the relay is

considered.

This paper is organized as follows. In Section II, the system

model is given. Then in Section III, a distributed version

of space-time codes where we employed in our system is

considered. Section IV discusses the PEP analysis in high

SNR scenarios and an optimum power allocation. Finally,

the overall performance comparisons for the system and the

proposed power allocation are presented in Section V.

I I . SYSTEM MODEL

Consider a network consisting of a source, a relay, anda destination node (Fig. 1). It is assumed that each node is

equipped with a single antenna.

Suppose each link has Rayleigh distribution and is inde-

pendent of the others. We denote the source-to-destination,

source-to-relay and relay-to-destination links by SD, SR,and RD, respectively. Both the destination and the relayare affected by zero-mean additive white complex Gaussian

noise with the variance of N0. In what follows, SD, SR,and RD links are demonstrated by hSD(n), hSR, and hRD,respectively. The variances of the related channel coefficients

are denoted by 2SD, 2

SR, and 2

RD. Path-loss in Rayleigh fading

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channel is equal to 1/d where is the path-loss exponent.Typically, lies in the range of 2 to 5 [14]. In contrast tospace-time codes in which transmit antennas usually send the

equal power, in distributed space-time codes for improving

the performance of the system, the available power is usually

allocated as a function of the relays position. Note that, the

proposed protocol in this paper exploits all degrees of freedomfor the decoding of the received signal. This implies that in

contrast to the proposed protocol in [4] in which just utilize the

transmitted information through the second phase, we employ

information radiated in both phases for the decoding purpose.

On the other hand, in contrast to the protocols in [3] in

which the source does not contribute in the second phase of

transmission, in our protocol the source contributes to transmit

data as well as relays in the second phase.

In this paper, the transmitted power in the first phase which

is radiated by the source is denoted by P1, and the transmittedpower in the second phase which is radiated by both the source

and relay is represented by P2. In the second phase, P2and (1 )P2 are transmitted by the source and the relay,respectively, where 0 1.

Since DF algorithm is used in this paper for transmitting

the data from the relay, it is necessary for the decoded data

in the relay to be almost without error. For achieving this

goal, transmitted power P1 is adjusting in such a way that theaverage received SNR in the relay, i.e., SNRR = P1/(

2

ndSR),

exceeds from a suitable threshold.

III. DISTRIBUTED DIFFERENTIAL SPACE-TIM E COD E

In this section, we describe the distributed differential space-

time code using the Alamouti structure. In this model, we

extend the differential space-time code proposed in [9] and

investigate its distributed version.The structure proposed in [9], when using the 2 2

Alamoutis scheme, can be shown as

S(0) = I2, S(n) = S(n 1)C(x1, x2), n = 1, 2, . . . (1)

where the 22 matrix S(0) is an arbitrary unitary matrix thatcould be the identity matrix I2. The matrix C(x1, x2) has theAlamouti structure and consists of symbols x1 and x2 chosenfrom M-PSK constellation. The time index n is skipped forsimplifying in representation of x1 and x2.

In the following, we explain the distributed form of the

differential structure described in (1). In this model, at the

first phase, the source S transmits the second row of the matrix

S(n) for R and D. In the second phase, S and R nodes transmitthe scaled rows of the matrix S(n) with appropriate powerratios, like two antennas belong to a node. We can express the

stated expressions as a following matrix form

SDDST(n) =

s21(n) s22(n) 1s11(n) 1s12(n)

0 0 2s21(n) 2s22(n)

,

(2)

where sij(n), i, j {1, 2} are the components ofS(n). Fur-thermore, two real coefficients 1 and 2 fulfill the equationP2 = P1(

2

1+2

2). These coefficients are introduced to have a

degree of freedom in distributing the power between S and R,

under a given total transmit power. The parameter introducedin Section I is related to 1 and 2 by = P1

2

1/P2.

Assuming r1(n) and r2(n) are the received signals in thefirst phase, and r3(n) and r4(n) are the received signals in thesecond phase. Stacking the signals received at the destination

terminal to form a 1 4 vector r(n), we obtain the following

input-output relation

r(n) =

P1h(n)SDDST(n) + w(n), (3)

where

r(n) =

r1(n) r2(n) r3(n) r4(n)

, (4)

h(n) =

hSD(n) hRD(n)

. (5)

The 1 4 vector w(n) consists of the zero-mean complexGaussian noise components with the covariance matrix of

E{w(n)Hw(n)} = N0I4, where (.)H stands for the complex

conjugate transpose and I4 denotes a 4 4 identity matrix.Using (1) and (2) we have

SDDST(n) = SDDST(n 1)(I2 C(n)), (6)

where is Kroncker product. Note that, since the matrix(I2 C(n)) is a unitary matrix, 1 and 2 in (2) are time-invariant parameters. If we regard (3) for two successive inter-

vals, and on the other hand, with assumption that the channel

characteristics remains unvarying, i.e., h(n) h(n1), using(6) we have

r(n) = r(n 1)(I2 C(n)) + w(n), (7)where

w(n) = w(n) w(n 1)(I2 C(n)). (8)It can be easily checked that w(n) is a zero-mean Gaus-sian noise with the covariance matrix of E{w(n)Hw(n)} =2N0I4, where noise variance is doubled in comparison withthe non-differential case. This is in conjunction with the con-

ventional 3dB performance loss in systems with the differential

detection.

IV. PERFORMANCE ANALYSIS AND POWER CONTROL

STRATEGY

For simplicity in expressing equations, we skip the time

index n from the matrix C(n). Thus, the coded matrix C(n)is shown by C1. If an error occurs in detecting C1, we

demonstrate it as C2.

An interesting property covered up in (7) is that we can

represent it as a non-differential structure with known channel

information where r(n 1) is equivalent to the channelcoefficients vector, and suppose the noise power is doubled.

On the other hand, the matrix (I2C(n)) is unitary. Thus, wecan employ pairwise error probability relationships in space-

time codes context [15]

P (C1 C2 | r(n 1))

= Q

Es

22bw

C2 C1 r(n 1)F

, (9)

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where .F is the Frobenius norm, and C1 and C2 are as

C1 = (I2 C1), (10)

C1 = (I2 C1), (11)

and 2bw is the noise variance in (7), and it is equal to 2N0. For

simplifying (9), we suppose high SNR conditions. Therefore,by neglecting the noise term in (3), we haveC2 C1 r(n 1)F

P1hSDDST(n 1)BSDDST(n 1)HhH, (12)

where

B =C1 C2

C1 C2

H. (13)

Now, we can normalize channel coefficients, or in other words,

separate the path-loss components from the fading channel.

Hence, (12) can be considered as

C2 C1

r(n 1)

F hAh

H, (14)

where, h is a 1 2 vector, and has zero-mean complexGaussian elements and unit variance. And,

A = P1LSDDST(n 1)BSDDST(n 1)HL, (15)

where,

L =

SD 0

0 RD

. (16)

Thus, PEP is given by [15]

P (C1 C2) =

E [P (C1 C2 | r(n 1))] 1

2

i=1 1 + Es

8N0i

, (17)

where i is the eigenvalues of the matrix A. Under high SNRscenarios, we have

P(C1 C2) 2i=1

Es

8N0i

1

=

Es

8N0

2

det1 (A) . (18)

where det (A) stands for the trace of the matrix A.Thus, theminimum of the probability of error is obtained by maximiz-

ing det (A)

maxdet (A) =

maxdet (B) det

P1LSDDST(n)SDDST(n)HL

max P1det (B) (dSDdRD) 1 + 2

12

2(19)

since

SDDST(n)SDDST(n)H =

diag|s21|

2 + |s22|2

+ 21

|s11|

2 + |s12|2

, 22

|s21|

2 + |s22|2

=

1 + 2

1

0

0 22

, (20)

where diag(.) denotes a diagonal matrix. One can verify (20)

considering (1) and the fact that multiplying two unitary

matrices become a unitary matrix. Therefore, for minimizing

the probability of error in high SNR scenarios, it is just enough

to maximize

1 + 21

22

, which leads to a simple power

control strategy that is independent of code and channel states.

As it is stated in Section II, the power P1 is chosen inorder to achieve a required average SNRR at the relay, defined

as SNRR = P1/(2

ndSR). As a result, P2, 1, and 2 can

be obtained by the following three equations, given the total

power in the network to be constant.1 + 2

1

= 2

2, (21)

P2 = P1

21 +

2

2

, (22)

P1 + P2 = P. (23)

Since = P12

1/P2, when P1 P2, we have

=1

2

P12P2

. (24)

It is important to note that this optimum point is independent

of the channel statistics and the position of the relay, which

improves its feasibility in the future communication systems.

When P1 P2, = 0 can be used. This means that allthe power in the second phase is transmitted by the relay

node. Although in this case the diversity of the second order is

achieved, but the usage of relay may decrease the performance

of the system. This is because the fact that the required P1for the errorless condition in the relay side could be high.

Thus, non-cooperative transmission may be preferable in this

case. Therefore, when P1 P2, in high SNR conditions,we can transmit all the power in the second phase from the

selected relay, and in low SNR conditions non-cooperative

transmission can be used. However, as we emphasis on high

SNR channels in our analysis, the amount of P1 which isobtained by the required SNR

Ris low enough in most of cases

to avoid P1 P2 case.

V. SIMULATION RESULTS

In this section, the performances of distributed differential

space-time codes are studied through simulations. The error

event is bit error rate (BER). In all simulations, 500 informa-

tion symbols are used in each frame. The fading is assumed to

be constant over each frame and vary independently from one

frame to another. The fading channels are flat with Rayleigh

distribution in which path-loss is also regarded. The path-

loss exponent is set to 2. The results are presented forQPSK. Moreover, for position of the nodes, we have supposed

dSD = 1 and dRD = 0.75. Furthermore, it is assumed thatthe relay decodes the received information without error. Thisassumption is easily achievable by reducing the SR link orincreasing the required SNRR.

Fig. 2 confirms the analytical results attained in Section IV

for finding the optimum of . In this figure, is defined as theratio of the transmitted power in Phase II to the transmitted

power in Phase I, i.e., = P2/P1. Then, substituting thevalues of equal to 1, 3 and 5 in (24) lead to the optimumvalues of as = 0, = 1/3, and = 0.4, respectively. Oneobserves in Fig. 2 that the optimum values of via analyticalresult lead to the optimum points in a sense of minimizing the

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Fig. 2. The average BER curves versus of a distributed differentialspace-time system with the employment of different detection with when thedestination SNR is 20 and dRD = 0.75.

BER by Monte-Carlo numerical simulations. This confirms the

correctness and usefulness of the brief equation of (24).

Fig. 3 illustrates the average bit error rate (BER) curve

as function of the average SNR per symbol for different

power allocations. Two different values of the parameter are employed. In each of them, the optimum power allocation

based on (24) is compared with the equal power allocation

to the source and the relay during the second phase, i.e.,

= 0.5. Note that the total transmitted power in networkis assumed to remain constant in all cases. Comparing two

curves with = 1 demonstrates a 2 dB gain in SNR at theBER of 103, when optimum power allocation is utilized.By increasing the value of to 2, the better performance isachieved. But, the performance gain due to using the opti-

mum power allocation becomes less significant, because the

optimum value of becomes close to 0.5. Note that the valueof is assigned by the required SNRR and considering thefact that the total transmitted power is constant. Furthermore,

Fig. 3 demonstrates that our proposed system outperforms the

distributed differential scheme proposed in [12] in DF mode.

Since one relay is utilized in our simulations, two differential

schemes introduced in [12] have the same structure.

VI . CONCLUSION

In this paper, a cooperative diversity scheme using dis-

tributed differential space-time codes in DF mode is proposed,

and examined. An outstanding characteristic of this scheme is

that CSI knowledge is not required in both the transmitters and

the receivers. Performance of the system is analyzed in high

SNR scenarios, and finally a simple and feasible power control

strategy is derived, which is independent of the relay position.

The performance of the system is investigated via numerical

simulations which demonstrate the proposed power allocation

outperforms a system with equal power transmission.

Fig. 3. The average BER performance of a distributed differential space-timesystem with the employment of different power allocations.

REFERENCES

[1] A. Sendonaris, E. Erkip and B. Aazhang, "User cooperation diversity.Part I. System description," IEEE Transactions on Communications, vol.51, no. 11, pp. 19271938, Nov. 2003.

[2] A. Sendonaris, E. Erkip and B. Aazhang, "User cooperation diversity. PartII. Implementation aspects and performance analysis," IEEE Transactionson Communications, vol. 51, no. 11, pp. 19391948, Nov. 2003.

[3] J. N. Laneman and G. W. Wornell, "Distributed space-time-coded pro-tocols for exploiting cooperative diversity in wireless networks," IEEETransactions on Information Theory, vol. 49, no. 10, pp. 24152425,Oct. 2003.

[4] Y. Jing and B. Hassibi, "Distributed space-time coding in wireless relaynetworks," in IEEE Sensor Array and Multichannel Signal ProcessingWorkshop, Sitges, Spain, July, 2004.

[5] R. U. Nabar, H. Bolcskei, and F. W. Kneubuhler, "Fading relay channels:

Performance limits and space-time signal design," IEEE Journal onSelected Areas in Communications, pp. 10991109, Aug. 2004.

[6] D. Chen and J.N. Laneman, "Cooperative diversity for wireless fadingchannels without channel state information," in Proc. Asilomar Conf.Signals, Systems, and Computers, Monterey, CA, Nov. 2004.

[7] B. L. Hughes, "Differential space-time modulation," IEEE Trans. Inform.Theory, vol. 46, pp. 25672578, Nov. 2000.

[8] B. M. Hochwald and T. L. Marzetta, "Unitary spacetime modulation formultiple-antenna communications in Rayleigh flat fading," IEEE Trans.

Inform. Theory, vol. 46, pp. 543-564, March 2000.[9] B. M. Hochwald and W. Sweldens, "Differential unitary space-time

modulation," IEEE Trans. Commun., vol. 48, pp. 20412052, Dec. 2000.[10] V. Tarokh and H. Jafarkhani, "A differential detection scheme for

transmit diversity," IEEE J. Select. Areas Commun., vol. 18, pp. 11691174, July 2000.

[11] B. Maham and F. Lahouti, "Cooperative communication using dis-tributed differential space-time codes," in IEEE International Confer-

ence on Wireless Communications, Networking and Mobile Computing(WiCOM 2006), Wuhan, China, Sep. 2224, 2006.

[12] W. Cho and L. Yang, "Distributed differential schemes for cooperativewireless networks," IEEE ICASSP, 2006.

[13] Q. Zhao, H. Li, "Decode-based differential modulation for wireless relaynetworks," IEEE ICASSP, Philadelphia, PA, March 2005.

[14] M. K. Simon and M.-S. Alouini, Digital Communication over FadingChannels: A Unified Approach to Performance Analysis. New York, NY:Wiley, 2000.

[15] H. Jafarkhani, Space-Time Coding Theory and Practice, CambridgeUniversity Press, 2005.